Math::PlanePath::UlamWarburton -- growth of a 2-D cellular automaton
use Math::PlanePath::UlamWarburton; my $path = Math::PlanePath::UlamWarburton->new; my ($x, $y) = $path->n_to_xy (123);
This is the pattern of a cellular automaton studied by Ulam and Warburton, numbering cells by growth tree row and anti-clockwise within the rows.
94 9 95 87 93 8 63 7 64 42 62 6 65 30 61 5 66 43 31 23 29 41 60 4 69 67 14 59 57 3 70 44 68 15 7 13 58 40 56 2 96 71 32 16 3 12 28 55 92 1 97 88 72 45 33 24 17 8 4 1 2 6 11 22 27 39 54 86 91 <- Y=0 98 73 34 18 5 10 26 53 90 -1 74 46 76 19 9 21 50 38 52 ... -2 75 77 20 85 51 -3 78 47 35 25 37 49 84 -4 79 36 83 -5 80 48 82 -6 81 -7 99 89 101 -8 100 -9 ^ -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
The growth rule is that a given cell grows up, down, left and right, but only if the new cell has no neighbours (up, down, left or right). So the initial cell "a" is N=1,
a initial depth=0 cell
The next row "b" cells are numbered N=2 to N=5 anti-clockwise from the right,
b b a b depth=1 b
Likewise the next row "c" cells N=6 to N=9. The "b" cells only grow outwards as 4 "c"s since the other positions would have neighbours in the existing "b"s.
c b c b a b c depth=2 b c
The "d" cells are then N=10 to N=21, numbered following the previous row "c" cell order and then anti-clockwise around each.
d d c d d b d d c b a b c d depth=3 d b d d c d d
There's only 4 "e" cells since among the "d"s only the X,Y axes won't have existing neighbours (the "b"s and "d"s).
e d d c d d b d e d c b a b c d e depth=4 d b d d c d d e
In general the pattern always grows by 1 outward along the X and Y axes and travels into the quarter planes between with a diamond shaped tree pattern which fills 11 of 16 cells in each 4x4 square block.
Counting depth=0 as the N=1 at the origin and depth=1 as the next N=2,3,4,5 generation, the number of cells in a row is
rowwidth(0) = 1 then rowwidth(depth) = 4 * 3^((count_1_bits(depth) - 1)
So depth=1 has 4*3^0=4 cells, as does depth=2 at N=6,7,8,9. Then depth=3 has 4*3^1=12 cells N=10 to N=21 because depth=3=0b11 has two 1-bits in binary. The N start and end for a row is the cumulative total of those before it,
Ndepth(depth) = 1 + (rowwidth(0) + ... + rowwidth(depth-1)) Nend(depth) = rowwidth(0) + ... + rowwidth(depth)
For example depth 3 ends at N=(1+4+4)=9.
depth Ndepth rowwidth Nend 0 1 1 1 1 2 4 5 2 6 4 9 3 10 12 21 4 22 4 25 5 26 12 37 6 38 12 49 7 50 36 85 8 86 4 89 9 90 12 101
For a power-of-2 depth the Ndepth is
Ndepth(2^a) = 2 + 4*(4^a-1)/3
For example depth=4=2^2 starts at N=2+4*(4^2-1)/3=22, or depth=8=2^3 starts N=2+4*(4^3-1)/3=86.
Further bits in the depth value contribute powers-of-4 with a tripling for each bit above. So if the depth number has bits a,b,c,d,etc in descending order,
depth = 2^a + 2^b + 2^c + 2^d ... a>b>c>d... Ndepth = 2 + 4*(-1 + 4^a + 3 * 4^b + 3^2 * 4^c + 3^3 * 4^d + ... ) / 3
For example depth=6 = 2^2+2^1 is Ndepth = 2 + (1+4*(4^2-1)/3) + 4^(1+1) = 38. Or depth=7 = 2^2+2^1+2^0 is Ndepth = 1 + (1+4*(4^2-1)/3) + 4^(1+1) + 3*4^(0+1) = 50.
The diamond shape depth=1 to depth=2^level-1 repeats three times. For example an "a" part going to the right of the origin "O",
d d d d | a d c --O a a a * c c c ... | a b c b b b b
The 2x2 diamond shaped "a" repeats pointing up, down and right as "b", "c" and "d". This resulting 4x4 diamond then likewise repeats up, down and right. The same happens in the other quarters of the plane.
The points in the path here are numbered by tree rows rather than in this sort of replication, but the replication helps to see the structure of the pattern.
Option parts => '2' confines the pattern to the upper half plane Y>=0,
parts => '2'
Y>=0
parts => "2" 28 6 21 5 29 22 16 20 27 4 11 3 30 12 6 10 26 2 23 13 3 9 19 1 31 24 17 14 7 4 1 2 5 8 15 18 25 <- Y=0 -------------------------------------- -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
Points are still numbered anti-clockwise around so X axis N=1,2,5,8,15,etc is the first of row depth=X. X negative axis N=1,4,7,14,etc is the last of row depth=-X. For depth=0 point N=1 is both the first and last of that row.
Within a row a line from point N to N+1 is always a 45-degree angle. This is true of each 3 direct children, but also across groups of children by symmetry. For this parts=2 the lines from the last of one row to the first of the next are horizontal, making an attractive pattern of diagonals and then across to the next row horizontally. For parts=4 or parts=1 the last to first lines are at various different slopes and so upsets the pattern.
Option parts => '1' confines the pattern to the first quadrant,
parts => '1'
parts => "1" to depth=14 14 | 73 13 | 63 12 | 53 62 72 11 | 49 10 | 39 48 71 9 | 35 47 61 8 | 31 34 38 46 52 60 70 7 | 29 45 59 6 | 19 28 69 67 5 | 15 27 57 4 | 11 14 18 26 68 58 51 56 66 3 | 9 25 23 43 2 | 5 8 24 17 22 44 37 42 65 1 | 3 7 13 21 33 41 55 Y=0 | 1 2 4 6 10 12 16 20 30 32 36 40 50 54 64 +----------------------------------------------- X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
X axis N=1,2,4,6,10,etc is the first of each row X=depth. Y axis N=1,3,5,9,11,etc is the last similarly Y=depth.
In this arrangement, horizontal arms have even, N and vertical arms have odd N. For example the vertical at X=8 N=30,33,37,etc has N odd from N=33 up and when it turns to horizontal at N=42 or N=56 it switches to N even. The children of N=66 are not shown but the verticals from there are N=79 below and N=81 above and so switch to odd again.
This odd/even pattern is true of N=2 horizontal and N=3 vertical and thereafter is true due to each row having an even number of points and the self-similar replications in the pattern,
|\ replication | \ block 0 to 1 and 3 |3 \ and block 0 block 2 less sides |---- |\ 2|\ | \ | \ |0 \|1 \ ---------
Block 0 is the base and is replicated as block 1 and in reverse as block 3. Block 2 is a further copy of block 0, but the two halves of block 0 rotated inward 90 degrees, so the X axis of block 0 becomes the vertical of block 2, and the Y axis of block 0 the horizontal of block 2. Those axis parts are dropped since they're already covered by block 1 and 3 and dropping them flips the odd/even parity to match the vertical/horizontal flip due to the 90-degree rotation.
Option parts => 'octant' confines the pattern to the first eighth of the plane 0<=Y<=X.
parts => 'octant'
parts => "octant" 7 | 47 ... 6 | 48 36 46 5 | 49 31 45 4 | 50 37 32 27 30 35 44 3 | 14 51 24 43 41 2 | 15 10 13 25 20 23 42 34 40 1 | 5 8 12 18 22 29 39 Y=0 | 1 2 3 4 6 7 9 11 16 17 19 21 26 28 33 38 +------------------------------------------------- X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
In this arrangement, N=1,2,3,4,6,7,etc on the X axis is the first N of each row (tree_depth_to_n()).
tree_depth_to_n()
Option parts => 'octant_up' confines the pattern to the upper octant 0<=X<=Y of the first quadrant.
parts => 'octant_up'
parts => "octant_up" 8 | 16 17 19 22 26 29 34 42 7 | 15 21 28 41 6 | 10 14 38 33 40 5 | 8 13 39 4 | 6 7 9 12 3 | 5 11 2 | 3 4 1 | 2 Y=0 | 1 +-------------------------- X=0 1 2 3 4 5 6 7
In this arrangement, N=1,2,3,5,6,8,etc on the Y axis the last N of each row (tree_depth_to_n_end()).
tree_depth_to_n_end()
The default is to number points starting N=1 as shown above. Option n_start can give a different start, in the same pattern. For example to start at 0,
n_start
n_start => 0 29 5 30 22 28 4 13 3 14 6 12 2 31 15 2 11 27 1 32 23 16 7 3 0 1 5 10 21 26 <- Y=0 33 17 4 9 25 -1 18 8 20 37 -2 19 -3 34 24 36 -4 35 -5 ^ -5 -4 -3 -2 -1 X=0 1 2 3 4 5
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::UlamWarburton->new ()
$path = Math::PlanePath::UlamWarburton->new (parts => $str, n_start => $n)
Create and return a new path object. The parts option (a string) can be
parts
"4" the default "2" "1"
@n_children = $path->tree_n_children($n)
Return the children of $n, or an empty list if $n has no children (including when $n < 1, ie. before the start of the path).
$n
$n < 1
The children are the cells turned on adjacent to $n at the next row. The way points are numbered means that when there's multiple children they're consecutive N values, for example at N=6 the children are 10,11,12.
@nums = $path->tree_num_children_list()
Return a list of the possible number of children in $path. This is the set of possible return values from tree_n_num_children(). The possible children varies with the parts,
$path
tree_n_num_children()
parts tree_num_children_list() ----- ------------------------ 4 0, 1, 3, 4 (the default) 2 0, 1, 2, 3 1 0, 1, 2, 3
parts=4 has 4 children at the origin N=0 and thereafter either 0, 1 or 3.
parts=2 and parts=1 can have 2 children on the boundaries where the 3rd child is chopped off, otherwise 0, 1 or 3.
$n_parent = $path->tree_n_parent($n)
Return the parent node of $n, or undef if $n <= 1 (the start of the path).
undef
$n <= 1
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return $n_lo = $n_start and
$n_lo = $n_start
parts $n_hi ----- ----- 4 $n_start + (16*4**$level - 4) / 3 2 $n_start + ( 8*4**$level - 5) / 3 + 2*2**$level 1 $n_start + ( 4*4**$level - 4) / 3 + 2*2**$level
$n_hi is tree_depth_to_n_end(2**($level+1) - 1.
$n_hi
tree_depth_to_n_end(2**($level+1) - 1
This cellular automaton is in Sloane's Online Encyclopedia of Integer Sequences as
http://oeis.org/A147562 (etc)
parts=4 A147562 total cells to depth, being tree_depth_to_n() n_start=0 A147582 added cells at depth A264039 off cells >=2 neighbours ("poisoned") A260490 increment A264768 off cells, 4 neighbours ("surrounded") A264769 increment parts=2 A183060 total cells to depth=n in half plane A183061 added cells at depth=n parts=1 A151922 total cells to depth=n in quadrant A079314 added cells at depth=n
The A147582 new cells sequence starts from n=1, so takes the innermost N=1 single cell as row n=1, then N=2,3,4,5 as row n=2 with 5 cells, etc. This makes the formula a binary 1-bits count on n-1 rather than on N the way rowwidth() above is expressed.
The 1-bits-count power 3^(count_1_bits(depth)) part of the rowwidth() is also separately in A048883, and as n-1 in A147610.
Math::PlanePath, Math::PlanePath::UlamWarburtonQuarter, Math::PlanePath::LCornerTree, Math::PlanePath::CellularRule
Math::PlanePath::SierpinskiTriangle (a similar binary 1s-count related calculation)
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
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